Analysis of spot patterns on a coordinate-invariant model for vegetation on a curved terrain

J. C. Tzou, L. Tzou

Research output: Contribution to journalArticlepeer-review


Motivated by the model proposed by Gandhi et al. in [J. R. Soc. Interface 15, 20180508],we propose a two-component reaction-advection-diffusion model for vegetation density and soil waterconcentration on a curved terrain with elevation given by z = ζ(x, y) and metric tensor g(x, y). It accountsfor downhill flow of soil water, spatially dependent effective evaporation of soil water, and vertical rainfallon a curved surface. In the singularly perturbed limit of slow vegetation diffusion 0 < ε2 << 1, weuse a hybrid asymptotic-numerical method to construct a localized quasi-equilibrium one-spot solutioncorresponding to one spot of a spotted vegetation pattern. We derive an ODE for the slow motion of thespot, finding that it is governed by three factors that arise at two different orders in ε. The leading order O(ε2| log ε|) effects are that of soil water advection and effective evaporation rate, the first driving the spotuphill while the second drive it toward regions of slow effective evaporation (e.g., valleys). Entering atO(ε2) are effects due to surface curvature along with higher order contributions of advection and variableevaporation rate. The combined effect of these three factors is encoded in the gradient of the regular partof a surface Green’s function for a second order linear operator of the form ∆g + v(x, y) · ∇ + V (x, y),for some velocity vector v(x, y) and effective potential V (x, y), where ∆g is the Laplace-Beltrami operatorcorresponding to metric tensor g(x, y). We compute this quantity using the analytic-numerical frameworkthat we first introduced in [Nonlinearity, 33, pp. 643-674], and demonstrate that it is critical for theaccurate prediction of spot motion. All numerical results are confirmed by finite element solutions of thefull PDE system
Original languageEnglish
Pages (from-to)2500-2529
Number of pages30
JournalSIAM Journal on Applied Dynamical Systems
Issue number4
Publication statusPublished - 2020


  • singularly perturbed reaction-advection-diffusion system
  • localized vegetation patterns
  • surface Green's function
  • matched asymptotic analysis
  • Hadamard parametrix
  • microlocal analysis

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